diff --git a/cf_xarray/accessor.py b/cf_xarray/accessor.py
index 08bc28ea..f747ba44 100644
--- a/cf_xarray/accessor.py
+++ b/cf_xarray/accessor.py
@@ -2727,7 +2727,9 @@ def grid_mapping_names(self) -> dict[str, list[str]]:
                 results[v].append(k)
         return results
 
-    def decode_vertical_coords(self, *, outnames=None, prefix=None):
+    def decode_vertical_coords(
+        self, *, outnames: dict[str, str] | None = None, prefix: str | None = None
+    ) -> None:
         """
         Decode parameterized vertical coordinates in place.
 
@@ -2765,6 +2767,7 @@ def decode_vertical_coords(self, *, outnames=None, prefix=None):
         allterms = self.formula_terms
 
         for dim in allterms:
+            dim = cast(str, dim)
             if prefix is None:
                 assert outnames is not None, (
                     "if prefix is None, outnames must be provided"
diff --git a/cf_xarray/parametric.py b/cf_xarray/parametric.py
index ca8a85ae..bd155329 100644
--- a/cf_xarray/parametric.py
+++ b/cf_xarray/parametric.py
@@ -782,7 +782,7 @@ def from_terms(cls, terms: dict):
         return cls(**get_terms(terms, "sigma", "depth", "z1", "z2", "a", "href", "k_c"))
 
 
-TRANSFORM_FROM_STDNAME = {
+TRANSFORM_FROM_STDNAME: dict[str, type[ParametricVerticalCoordinate]] = {
     "atmosphere_ln_pressure_coordinate": AtmosphereLnPressure,
     "atmosphere_sigma_coordinate": AtmosphereSigma,
     "atmosphere_hybrid_sigma_pressure_coordinate": AtmosphereHybridSigmaPressure,